Theorem opprvalg 13701

Description: Value of the opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)

Hypotheses

Ref	Expression
opprval.1	|- B = ( Base ` R )
opprval.2	|- .x. = ( .r ` R )
opprval.3	|- O = (oppr ` R )

Assertion

Ref	Expression
opprvalg	|- ( R e. V -> O = ( R sSet <. ( .r ` ndx ) , tpos .x. >. ) )

Proof of Theorem opprvalg

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opprval.3	. 2 |- O = (oppr ` R )
2		df-oppr 13700	. . 3 |- oppr = ( x e. _V |-> ( x sSet <. ( .r ` ndx ) , tpos ( .r ` x ) >. ) )
3		id 19	. . . 4 |- ( x = R -> x = R )
4		fveq2 5561	. . . . . . 7 |- ( x = R -> ( .r ` x ) = ( .r ` R ) )
5		opprval.2	. . . . . . 7 |- .x. = ( .r ` R )
6	4, 5	eqtr4di 2247	. . . . . 6 |- ( x = R -> ( .r ` x ) = .x. )
7	6	tposeqd 6315	. . . . 5 |- ( x = R -> tpos ( .r ` x ) = tpos .x. )
8	7	opeq2d 3816	. . . 4 |- ( x = R -> <. ( .r ` ndx ) , tpos ( .r ` x ) >. = <. ( .r ` ndx ) , tpos .x. >. )
9	3, 8	oveq12d 5943	. . 3 |- ( x = R -> ( x sSet <. ( .r ` ndx ) , tpos ( .r ` x ) >. ) = ( R sSet <. ( .r ` ndx ) , tpos .x. >. ) )
10		elex 2774	. . 3 |- ( R e. V -> R e. _V )
11		mulrslid 12834	. . . . . 6 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
12	11	simpri 113	. . . . 5 |- ( .r ` ndx ) e. NN
13	12	a1i 9	. . . 4 |- ( R e. V -> ( .r ` ndx ) e. NN )
14	11	slotex 12730	. . . . . 6 |- ( R e. V -> ( .r ` R ) e. _V )
15	5, 14	eqeltrid 2283	. . . . 5 |- ( R e. V -> .x. e. _V )
16		tposexg 6325	. . . . 5 |- ( .x. e. _V -> tpos .x. e. _V )
17	15, 16	syl 14	. . . 4 |- ( R e. V -> tpos .x. e. _V )
18		setsex 12735	. . . 4 |- ( ( R e. _V /\ ( .r ` ndx ) e. NN /\ tpos .x. e. _V ) -> ( R sSet <. ( .r ` ndx ) , tpos .x. >. ) e. _V )
19	10, 13, 17, 18	syl3anc 1249	. . 3 |- ( R e. V -> ( R sSet <. ( .r ` ndx ) , tpos .x. >. ) e. _V )
20	2, 9, 10, 19	fvmptd3 5658	. 2 |- ( R e. V -> (oppr ` R ) = ( R sSet <. ( .r ` ndx ) , tpos .x. >. ) )
21	1, 20	eqtrid 2241	1 |- ( R e. V -> O = ( R sSet <. ( .r ` ndx ) , tpos .x. >. ) )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2167   _Vcvv 2763   <.cop 3626   ` cfv 5259  (class class class)co 5925  tpos ctpos 6311   NNcn 9007   ndxcnx 12700   sSet csts 12701  Slot cslot 12702   Basecbs 12703   .rcmulr 12781  opp_rcoppr 13699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1re 7990  ax-addrcl 7993

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-tpos 6312  df-inn 9008  df-2 9066  df-3 9067  df-ndx 12706  df-slot 12707  df-sets 12710  df-mulr 12794  df-oppr 13700

This theorem is referenced by:  opprmulfvalg  13702  opprex  13705  opprsllem  13706